Everyday Usage and usage in mathematics

In everyday language, the word correlate is used to describe a relationship or connection between two things. For example, we might say "The amount of sleep you get correlates with your overall health," meaning that there is a connection between the amount of sleep and your health where changes in one are associated with changes in the other.

The term correlation coefficient is used in mathematics with a technical meaning that is similar to the everyday use of the word correlation. denoted by \( r \), is a numerical value that indicates the strength and direction of a linear relationship between two variables.

Importance in mathematics

Linear Relationships: The correlation coefficient is meaningful primarily when the data exhibits a linear relationship. For instance, if you're plotting data points on a graph and they lie close to a straight line, the correlation coefficient tells you how well the data points fit that line.

Values of \( r \): The correlation coefficient can range from \( -1 \) to \( 1 \):

• \( r > 0 \) suggests a positive relationship, meaning as one variable increases, so does the other.
• \( r < 0 \) suggests a negative relationship, meaning as one variable increases, the other decreases.
• The closer \( r \) is to \( 0 \), the more scattered the data points are, indicating a weaker linear relationship.
• The closer \( r \) is to \( 1 \) or \( -1 \), the less scattered the data points are, indicating a stronger linear relationship.

Interpreting \( r \): If \( r \) is close to \( 1 \) or \( -1 \), the data closely follows a straight line, meaning the relationship between the variables is strong and predictable. Conversely, if \( r \) is near \( 0 \), the data points are more dispersed, and the relationship between the variables is weak.

Exercises

• Does your native language have a word that means the same thing as correlation? Try to describe the meaning of the English word correlation in your native language.
• If $t$ is the time spent studying for an exam and $s$ is the score on the exam, would you expect the correlation coefficient between the variables $s$ and $t$ to be positive or negative?